Author 
Topic: Prime pairs is NOT hard (Read 2407 times) 

Dmitriy
Guest

Guys, this one is the simplest of them all. 1. Out of 3 successive numbers one can always be divided by 3. The proof is elementary. 2. Each second number is even. 3. Primes can only be divided by themselves. Thus the number between them can be divided by both 2 and 3, which means it can be divided by 6.


IP Logged 



Gamer555
Newbie
Posts: 19


Re: Prime pairs is NOT hard
« Reply #1 on: Jul 30^{th}, 2002, 9:10am » 
Quote Modify

That's a good proof, lots simpler than the one I had. What I would say is: (EXACTLY what you said, am not taking credit for what you said) All primes above 6 odd. So the number between them is even, and divisible by *2* Out of 3 numbers, one must be divisible by three, and since all prime numbers aren't divisible by three, the one in the middle must be. Is this simpler to read?


IP Logged 



NickH
Senior Riddler
Gender:
Posts: 341


Re: Prime pairs is NOT hard
« Reply #2 on: Jul 30^{th}, 2002, 10:20am » 
Quote Modify

The above are all good arguments. An equivalent approach is to observe that all primes greater than 3 are of the form 6n1 or 6n+1. (6n, 6n+2, 6n2, 6n+3 are clearly composite.) Thus prime paris must be 6k1 and 6k+1, for some k, and so the number in the middle is 6k.


IP Logged 
Nick's Mathematical Puzzles



anshil
Newbie
Posts: 41


Re: Prime pairs is NOT hard
« Reply #3 on: Aug 1^{st}, 2002, 12:57am » 
Quote Modify

Okay other question, how many prime pairs are there? If you guess infinite, can you mathematically proof that there are infinite number of prime pairs? ( Similar like there exists a proof that there are infinite number of primes )

« Last Edit: Aug 1^{st}, 2002, 12:57am by anshil » 
IP Logged 



NickH
Senior Riddler
Gender:
Posts: 341


Re: Prime pairs is NOT hard
« Reply #4 on: Aug 2^{nd}, 2002, 11:45am » 
Quote Modify

As I recall, it is an open question whether the number of prime pairs is finite or infinite. It IS known that, whereas the sum of the reciprocals of the primes is infinite, the sum of the reciprocals of all prime pairs is finite. In fact, it is less than 2.


IP Logged 
Nick's Mathematical Puzzles



